I would suggest making use of application containers to simplify your application structure. Slim 3 has been designed to work well with application containers.

Pass the container to your class method - you will then have the request and response objects available via the (shared) container, since Slim assigns those (request and response) to the container object automatically.

You can even add/assign your database connection (and whatever else you want to make available to other classes) to the container, then you only need to pass the same container to all functions that require database functionality.

The idea is that you can write classes that can be re-used in other projects, even if you decide to use something different than Slim next time. As long as the framework uses application containers, you can probably re-use your classes.